Questions tagged [young-tableaux]

For standard Young tableaux, semistandard Young tableaux, and other related two-dimensional arrays of numbers like plane partitions. Including their combinatorial theory and their application in representation theory and algebraic geometry.

Filter by
Sorted by
Tagged with
2 votes
0 answers
31 views

Why are the terms in this sequence of Littlewood-Richardson coefficients congruent to 1 mod 8?

Let $n$ be a positive integer and write $n=q_1+q_2+\dots+q_t$ as a sum of distinct powers of $2$. Consider the partition $\lambda(n)$ of $2n$ whose Frobenius symbol is $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{...
user avatar
9 votes
0 answers
168 views

Hives for other root systems?

Littlewood-Richardson coefficients $c^\lambda_{\mu\nu}$ have numerous combinatorial interpretations, including the hive model by Knutson and Tao, see here. For other root systems, there are also ...
user avatar
  • 15.2k
4 votes
0 answers
144 views

Relationship between $\mathbb{S}^{\nu}V \otimes \mathbb{S}^{\lambda}(V^{*})$ and $\mathbb{S}^{\nu / \lambda}V$

For partition $\mu$ let $\mathbb{S}^{\mu}V = V^{\otimes \mu} \cdot c_{\mu}$, where $c_{\mu}$ is the Young symmetrizer. I'm trying to prove that $\mathbb{S}^{\nu / \lambda}V$ is the polynomial part of $...
user avatar
  • 49
0 votes
0 answers
67 views

Addition theorem for Schur function in multivariable

Working with the following problem Expansion in Schur function of negative binomial exponent I need to find the expansion of $$ s_{\lambda}(x_1 + y , x_2 +y, \ldots, x_n +y)$$ in terms of schur ...
user avatar
  • 621
2 votes
0 answers
71 views

Yamanouchi ribbon tableaux?

Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions. The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ ...
user avatar
  • 643
1 vote
0 answers
122 views

Status of conjecture of Conrey and Gonek, combinatorial meaning

I was looking at the OEIS on the number of square Young Tableaux. In it Michael Somos referenced a paper of Conrey and Gonek, High Moment's of the Riemann Zeta-Function. Is there an combinatorial ...
user avatar
  • 741
1 vote
0 answers
61 views

Scalars by which symmetrizations of cyclic permutations act on Specht modules

Let $S_n$ be the symmetric group. Pick $a \in 2,\ldots,n$ and denote by $c_a \in \mathbb{C}[S_n]$ the symmetrization of the element $(12\ldots a)$ i.e. $c_a$ is the sum of cycles of type $a$. Let $\...
user avatar
  • 123
1 vote
1 answer
112 views

Number of paths to a specific vertex in the Young's lattice

Consider the Young's lattice. What is the number of paths starting from the origin (0) to a specific Young diagram? For instance, the Young diagram corresponding to the integer partition 1+1+1 has 1 ...
user avatar
2 votes
1 answer
113 views

Number of branches between two layers of the Young's lattice

In the Young's lattice, the number of branches that connect the $N$'th layer to the $N+1$'th layer has the sequence: $$ 1,2, 4, 7, 12, 19, 30, 45, 67, 97, 139, \cdots $$ Looking this up on OEIS, leads ...
user avatar
11 votes
1 answer
236 views

Lattice structure (wrt dominance order) on the set of Young diagrams appearing in the decompositions given by the Littlewood-Richardson rule

The irreducible decomposition of the tensor product of two irreducible representations of GL(n) is described by the Littlewood-Richardson rule. This same rule also governs the decomposition of the ...
user avatar
4 votes
0 answers
113 views

Map between irreducible representations in basis given by Young tableaux

Let $V$ be a $n$-dimensional complex vector space. Assume we have a $\mathbb C$-linear map $\varphi:\Gamma^{(a_1,\dots,a_n)}V\rightarrow \Gamma^{(b_1,\dots,b_n)}V$ between two irreducible ...
user avatar
  • 1,351
1 vote
0 answers
66 views

LGV scheme: Any situations where the weights shift differently for each path?

In Cylindric partitions, Proposition 1, Gessel and Krattenthaler prove a formula for lattice paths on a cylinder In our particular problem, we again have paths $((P_{1},k_{1}),...,(P_{r},k_{r}))$ but ...
user avatar
  • 1,378
9 votes
0 answers
448 views

Two majs for standard Young tableaux?

Let $\lambda$ be a partition of $n$, and consider its set of standard Young tableaux (SYTs): bijective fillings of the Young diagram of $\lambda$, written in English notation, with the numbers $1$ ...
user avatar
  • 18.8k
4 votes
2 answers
252 views

LGV scheme for lattice paths that move in non-unit spatial positive steps

In the Lindström–Gessel–Viennot lemma (LGV) applied to the $Z^2$-lattice paths are taken to move in unit spatial-steps in unit time (see here). What do we mean by "time"? In the language of ...
user avatar
  • 1,378
8 votes
0 answers
222 views

A question related to Young symmetrizers

Let $T$ be an arbitrary Young tableau (i.e., filling of the diagram of an integer partition $\lambda$ of $n$ by the numbers from $1$ to $n$, each appearing once). Let $R(T)$ be the subgroup of ...
user avatar
2 votes
0 answers
79 views

The distribution of amajor over standard Young tableaux?

Given a standard Young tableau $T$, $i$ is called an ascent of $T$ if $i+1$ is in a higher row than $i$, and amajor$(T)$ is defined to be the sum of ascents of $T$. For the distribution of the amajor ...
user avatar
  • 115
1 vote
0 answers
89 views

RSK correspondence for sum of two matrices

The celebrated RSK correspondence (see Wikipedia page) assigns to each integer $X$ matrix a pair of Young tableaux $P$ and $Q$. Now suppose that we have three integer matrices $X_1$, $X_2$, and $X_3$, ...
user avatar
  • 121
5 votes
1 answer
428 views

A basic question about Young symmetrizers

This is probably elementary for experts on the representation theory of the symmetric group, but I did not find the answers I need by a cursory look at the usual textbooks (they could be there, but I ...
user avatar
8 votes
0 answers
159 views

Generalized Young symmetrizers, why not?

For $n$ a positive integer, let $[n]=\{1,2,\ldots,n\}$. Consider two set partitions $\mathcal{A}=\{A_1,\ldots,A_p\}$ and $\mathcal{B}=\{B_1,\ldots,B_q\}$ of the set $[n]$. We will denote by $G(\...
user avatar
7 votes
0 answers
94 views

A question regarding an analog of Young symmetrizer: the product row and column preserving subgroups without sign representation

Consider a rectangular Young diagram $\lambda$ with $n = pq$ boxes, with $p$ rows and $q$ columns. If $C$ is the column preserving subgroup of $\lambda$ and $R$ is the row preserving subgroup, then we ...
user avatar
  • 121
7 votes
1 answer
465 views

Bender-Knuth involutions for symplectic (King) tableaux

First let me recall the combinatorial theory of the characters of $\mathfrak{gl}_m$, a.k.a., Schur polynomials. For a partition $\lambda$, a semistandard Young tableaux of shape $\lambda$ is a filling ...
user avatar
  • 18.8k
3 votes
0 answers
113 views

Consequences of Littlewood-Richardson rule

I am trying to read Deligne's paper 'Categories Tensorielles', and in the first chapter Deligne states some results obtained from the Littlewood-Richardson rule that I do not understand. He states: '...
user avatar
  • 275
2 votes
0 answers
47 views

Expansion of polytabloids in the standard basis

would like to know the most efficient way to write a polytabloid in terms of standard ones. I know the Garnir elements, but using them to do calculations is hard. I also read about "quadratic ...
user avatar
  • 643
6 votes
1 answer
419 views

Refined reverse plane partition generating function

I have a simple question about the generating function for reverse plane partitions: $$\sum_{\pi \in RPP(\lambda)} z^{|\pi|}= \prod_{s \in \lambda} \frac{1}{1-z^{h_{\lambda}(s)}}$$ There's a ...
user avatar
8 votes
1 answer
342 views

RSK correspondence

Up to now, what are the difference ways we know to define RSK correspondence? I already know: By insertion and recording tableau. Ball construction or Viennot's geometric construction. Growth diagram ...
user avatar
  • 320
16 votes
3 answers
650 views

Plane partitions with equal margins

A plane partition of $n$ is an table of integers $A=(a_{ij})$ which add up to $n$ and non-increase in rows and columns. For example, $$A= \begin{matrix} 331 \\ 32 \ \ \\ 11 \ \ \end{matrix} $$ is a ...
user avatar
  • 15.2k
3 votes
0 answers
111 views

Tableaux switching

I'm reading the article Tableau Switching: Algorithms and Applications by Benkart, Sottile, and Stroomer. Do you know if there are any articles or books that talk more about the properties of tableau ...
user avatar
  • 320
5 votes
0 answers
281 views

Determinantal formula for plane partitions of shifted shape

For $\lambda = (\lambda_1,\ldots,\lambda_{\ell})$ a partition, a (weak) plane partition of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with nonnegative integers such that entries ...
user avatar
  • 18.8k
5 votes
3 answers
272 views

Tensoring irreducible $B$-series representations/ Type B Littlewood-Richardson

When tensoring finite dimensional representations of the Lie algebra ${\frak sl}_n$, we have an explicit algorithm given in terms of Young diagrams. See Section 4 of this paper. Do there exist ...
user avatar
2 votes
0 answers
87 views

Restricted Cauchy identity

Is there some reference for sums like: $$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)t^{|\nu|}$$ $$\sum_{\nu \subset \mathrm{[1,n] x[1, m]}}s_{\nu}(x)s_{\nu}(y)\cdot|\nu|$$ (summation ...
user avatar
3 votes
0 answers
174 views

Constructing a centrally primitive idempotent in the group algebra of the symmetric group

Consider the group algebra of the symmetric group $ \mathbb{C} S_k$. Given some Young tableau $T$ of shape $\lambda$, let $a_{\lambda,T}$ and $b_{\lambda,T}$ be the row symmetrizer and column ...
user avatar
5 votes
1 answer
333 views

Iterated derivative and rectangular standard Young tableaux

We first make a few definitions, seemingly out of the blue (they are introduced/defined in this paper). Let $F^0_{a}(z) = (1-z)^{-1}$ and define recursively $$ F^{k+1}_{a}(z) = z^{a-1} \frac{d^a}{dz^...
user avatar
4 votes
1 answer
172 views

Decomposing tensor powers of the fundamental representation of exceptional Lie algebras

For the $A$-series, tensor powers of the fundamental representation of $\frak{sl}_n$ decompose into irreducibles according to a certain Young diagram/ partition formula. This inspires, for example, ...
user avatar
10 votes
1 answer
292 views

About $K$-rectification of increasing tableaux

Let $T$ be a standard Young tableaux on $[n]$. Denote the RSK algorithm $\text{RSK}(w)=(P(T),Q(T))$ for $w\in\mathfrak{S}_n$, where $P(T)$ is the Schencted insertion tableaux. For $1\leq i\leq j\leq n$...
user avatar
4 votes
0 answers
96 views

Intersection of components in Springer fibre of type A

From the standard results on Springer fibers of type A, we know that given a Springer fiber, say $\mathcal{B}_\lambda,$ its irreducible components are all equidimensional and parametrized by standard ...
user avatar
  • 1,429
13 votes
1 answer
511 views

Coincidences between average Catalan tableaux

There are Catalan number $C_n$ of standard Young tableaux of shape $(n,n)$, which we view as $2\times n$ matrices. Denote by $P_n$ the average of these matrices: $$ P_n \, := \, \frac{1}{C_n} \, \...
user avatar
  • 15.2k
8 votes
0 answers
127 views

Continuous analogues of Schützenberger promotion

Has anyone studied continuous analogues of Schützenberger promotion, and in particular, a flow on (a suitable subset of) the order polytope of a poset? Here’s what I have in mind: Given a poset $P$, ...
user avatar
  • 17.2k
2 votes
1 answer
201 views

Major index generating polynomial for border-strip tableaux

The Question in its original form has been answered, but there is a follow-up, see the end of the post. A border-strip is a skew Young diagram that does not contain a $2 \times 2$-box. A border-strip ...
user avatar
5 votes
1 answer
207 views

Generating function for lattice paths making aribitrary (i,j)-up-right move in one step and fitting rectangular (m,n)?

There is the following beautiful formula (see Qiaochu Yuan excellent blog): $$ \sum_{\lambda \in Young~diagrams~fitting~rectangle~m~n} q^{Box~count(="area~under~the~curve")~of~\lambda} = \binom{n+m}{...
user avatar
1 vote
0 answers
55 views

To find $\dim(D^{\mu})=\dim\frac{S^{\mu}}{S^{\mu}\cap S^{\mu\perp}}$

Let $S_6$ be the symmetric group of degree $6$ and $F$ be any finite field of characteristic $2.$ Then $2$-regular partition of $6$ are $(5,1)$, $(4,2)$ and $(3,2,1)$ . I have to find $$\dim(D^{\mu})=\...
user avatar
  • 121
6 votes
1 answer
233 views

Schutzenberger's evacuation and $\mu$-coefficient of Kazhdan–Lusztig polynomials

$\def\SYT{\mathrm{SYT}}\def\RSK{\mathrm{RSK}}\DeclareMathOperator\evac{evac}$Let $\mathfrak{S}_n$ be the symmetric group, $\SYT_n$ be the set of standard young tableaux of size $n$. For $u\in \...
user avatar
6 votes
0 answers
136 views

How to represent the even signed permutations by Young tableaux?

The well-known RSK correspondence established the connection between table pair (P,Q) and the permutations in symmetry group Sn(Coxeter group of type A). Also, there is a similar correspondence for ...
user avatar
  • 61
4 votes
1 answer
318 views

Bijection between noncrossing matchings on $2b$ points and Standard Young Tableaux of size $2 \times b$

I'm currently reading a review article called Dynamical Algebraic Combinatorics: Promotion, Rowmotion, and Resonance by Jessica Striker. In this article, Striker writes that there is a ''nice'' ...
user avatar
3 votes
1 answer
221 views

Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$?

Nakajima & Yoshioka [1] showed that \begin{equation} F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q}) = \sum_{n = 1}^\infty \mathbf{q}^nF^{inst}_n(\epsilon_1,\epsilon_2,\mathbf{a}) := \...
user avatar
24 votes
3 answers
1k views

Is the Ford-Fulkerson algorithm a tropical rational function?

The Ford-Fulkerson algorithm Let me recall the standard scenario of flow optimization (for integer flows at least): Let $\mathbb{N} = \left\{0,1,2,\ldots\right\}$. Consider a digraph $D$ with vertex ...
user avatar
1 vote
0 answers
58 views

A dimension formula for generalised symmetric powers of the natural module

I need a reference for the following well-known statement - does anyone know one? Let $\mu$ a partition of $n$ into at most $d$ parts. We let $${\rm Sym}^\mu(\Bbbk^d)={\rm Sym}^{\mu_1}(\Bbbk^d) \...
user avatar
  • 1,171
5 votes
0 answers
222 views

Tabloid Construction of Permutation Representation of Hyperoctahedral Group

For a partition $\lambda \vdash n$, the permutation representation $M^{\lambda}$ of the symmetric group can be constructed in two ways. First, it may be written as the induced representation $M^{\...
user avatar
3 votes
0 answers
175 views

Orthogonal basis for decomposition of induced representation of derangements

Background Let $V_n$ be the $\mathbb{C}$-module spanned by the set of derangements (permutations with no fixed points) inside the group ring of $S_n$. We make $V_n$ into a $\mathbb{C}S_n$-module ...
user avatar
0 votes
1 answer
274 views

Dimension of irreducible representation associated to a Young tableau

This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here. Suppose that $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k$ and $\rho$ be ...
user avatar
1 vote
1 answer
187 views

Skew character with hooks

I have asked this in MSE 8 days ago, even offered a bounty, and got nothing, so will try here. I would like to understand the value of the skew characters of the symmetric group, $\chi_{\lambda/\mu}$ ...
user avatar
  • 1,033